Open Access
This article is

- freely available
- re-usable

*Universe*
**2019**,
*5*(9),
200;
https://doi.org/10.3390/universe5090200

Article

Gauss–Bonnet Inflation and the String Swampland

^{1}

Department of Astronomy, Beijing Normal University, Beijing 100087, China

^{2}

School of Physics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China

^{*}

Author to whom correspondence should be addressed.

Received: 22 August 2019 / Accepted: 12 September 2019 / Published: 15 September 2019

## Abstract

**:**

The swampland criteria are generically in tension with single-field slow-roll inflation because the first swampland criterion requires small tensor-to-scalar ratio while the second swampland criterion requires either large tensor-to-scalar ratio or large scalar spectral tilt. The challenge to single-field slow-roll inflation imposed by the swampland criteria can be avoided by modifying the relationship between the tensor-to-scalar ratio and the slow-roll parameter. We show that the Gauss–Bonnet inflation with the coupling function inversely proportional to the potential overcomes the challenge by adding a constant factor in the relationship between the tensor-to-scalar ratio and the slow-roll parameter. For the Gauss–Bonnet inflation, while the swampland criteria are satisfied, the slow-roll conditions are also fulfilled, so the scalar spectral tilt and the tensor-to-scalar ratio are consistent with the observations. We use the potentials for chaotic inflation and the E-model as examples to show that the models pass all the constraints. The Gauss–Bonnet coupling seems a way out of the swampland issue for single-field inflationary models.

Keywords:

inflation; Gauss–Bonnet inflation; swampland## 1. Introduction

Inflation solves the flatness and horizon problems in standard cosmology [1,2,3,4,5], and is usually modeled by a single slow-roll scalar field which is obtained from low-energy effective field theories. In order to embed such scalar fields in a string quantum gravity theory successfully, they have to satisfy the following swampland criteria [6,7]:

- Swampland Criterion I ($\mathcal{SC}$I) [8]: The scalar field excursion, normalized by the reduced Planck mass, in field space is bounded from above$$|\Delta \varphi |\le d,$$
- Refined de Sitter Conjecture ($\mathcal{SC}$II) [9,10]: The gradient of the field potential V with $V>0$ should satisfy$$\frac{|\nabla V|}{V}\ge c,$$$$\frac{\mathrm{min}\left({\nabla}_{i}{\nabla}_{j}V\right)}{V}\le -\tilde{c},$$

For single-field slow-roll inflation, the first two slow-roll parameters are ${\u03f5}_{V}={({V}^{\prime}/V)}^{2}/2$ and ${\eta}_{V}={V}^{\u2033}/V$, and the tensor-to-scalar ratio is $r=16{\u03f5}_{V}$, where ${V}^{\prime}=dV/d\varphi $. In terms of the slow-roll parameters, condition (2) becomes ${\u03f5}_{V}\ge {c}^{2}/2$ and condition (3) becomes ${\eta}_{V}\le -\tilde{c}$. Obviously, condition (2) violates the slow-roll condition and poses a threat to inflationary models by requiring a large tensor-to-scalar ratio $r\sim 8{c}^{2}$. For example, even if we chose $c=0.1$ [11], it is still inconsistent with the observational constraint ${r}_{0.002}<0.064$ [12,13] because $r\sim 0.08>0.064$. Condition (3) also poses a threat to single-field slow-roll inflation because it requires that ${n}_{s}=1+2{\eta}_{V}-3r/8<1-2\tilde{c}\sim -\mathcal{O}\left(1\right)$, which is inconsistent with the observation. As pointed out in Reference [14], a viable way to solve this problem is by using models with the tensor-to-scalar ratio r reduced by a factor while keeping the lower bound on the field excursion $\Delta \varphi $ as required by the Lyth bound [15], such as warm inflation [16]. See References [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48] for more discussions on this issue.

In Reference [49], we find a powerful mechanism to reduce the tensor-to-scalar ratio r. With the help of the Gauss–Bonnet coupling, for any potential, the tensor-to-scalar r is reduced by a factor of ${(1-\lambda )}^{2}$ with the order one parameter $\lambda $, so it may solve the swampland problem. Furthermore, the Gauss–Bonnet term is induced from the superstring theory, and it may solve the singularity problem of the Universe [50,51,52,53,54]. The speed of gravitational waves in Gauss–Bonnet inflation is usually different from the speed of light, ${c}_{T}\ne 1$. However, as discussed in [55], the effect of the Gauss–Bonnet term is negligible at low energy, and the Gauss–Bonnet model is compatible with the observational constraint ${c}_{T}\approx 1$ [56]. Furthermore, it was found that by choosing the coupling function appropriately we can keep ${c}_{T}=1$ in Gauss–Bonnet inflation [57].

Generally, the predictions of the inflation, ${n}_{s}$ and r, are calculated under the slow-roll conditions ${\u03f5}_{V}\ll 1$ and $|{\eta}_{V}|\le 1$. If the $\mathcal{SC}$II criterion (2) or (3) is satisfied, then the slow-roll condition will be violated, and the predictions ${n}_{s}$ and r may be unreliable. However, in the case with Gauss–Bonnet coupling, the slow-roll condition is $(1-\lambda ){({V}^{\prime}/V)}^{2}\ll 1$, so even the second swampland criterion (2) is satisfied—as long as $1-\lambda \ll 1$, the model still satisfies the slow-roll condition, so the slow-roll results are applicable. In this paper, we show that with the help of the Gauss–Bonnet coupling, some inflationary models satisfy not only the swampland criteria but also the observational constraints.

This paper is organized as follows. In Section 2, we give a brief introduction to the Gauss–Bonnet inflation, and point out the reason why it is easy to satisfy the swampland criteria. In Section 3, we use the power-law potential and the E-model to show that all the constraints can be satisfied. We conclude the paper in Section 4.

## 2. The Gauss–Bonnet Inflation

The action for Gauss–Bonnet inflation is [58,59,60,61,62,63,64,65,66]
where ${R}_{\mathrm{GB}}^{2}={R}_{\mu \nu \rho \sigma}{R}^{\mu \nu \rho \sigma}-4{R}_{\mu \nu}{R}^{\mu \nu}+{R}^{2}$ is the Gauss–Bonnet term which is a pure topological term in four dimensions, and $\xi \left(\varphi \right)$ is the Gauss–Bonnet coupling function. For a more detailed discussion on Gauss–Bonnet inflation, please refer to [67] and references therein. In this paper, we use [49,68]
where $0<\lambda <1$. The parameter ${\mathsf{\Lambda}}_{0}\ll {\left({10}^{16}\mathrm{Gev}\right)}^{4}$ added here is to avoid the reheating problem of Gauss–Bonnet inflation [68], and it can be ignored during inflation, so in this paper we neglect the effect of ${\mathsf{\Lambda}}_{0}$. In terms of the horizon flow slow-roll parameters [69], the slow-roll conditions are
the e-folding number N at the horizon exit before the end of the inflation is
the scalar spectral tilt ${n}_{s}$ and the tensor-to-scalar ratio r are [49]

$$S=\frac{1}{2}\int \sqrt{-g}{d}^{4}x\left[R-{g}^{\mu \nu}{\partial}_{\mu}\varphi {\partial}_{\nu}\varphi -2V\left(\varphi \right)-\xi \left(\varphi \right){R}_{GB}^{2}\right],$$

$$\xi \left(\varphi \right)=\frac{3\lambda}{4V\left(\varphi \right)+{\mathsf{\Lambda}}_{0}},$$

$$\begin{array}{c}{\u03f5}_{1}=-\frac{\dot{H}}{{H}^{2}}\ll 1,\phantom{\rule{1.em}{0ex}}{\u03f5}_{2}=\frac{\dot{{\u03f5}_{1}}}{H{\u03f5}_{1}}\ll 1,\end{array}$$

$$N={\int}_{{\varphi}_{e}}^{\varphi}\frac{1}{\sqrt{2{\u03f5}_{1}}}d\varphi ,$$

$$\begin{array}{c}{n}_{s}-1=-2{\u03f5}_{1}-{\u03f5}_{2},\end{array}$$

$$\begin{array}{c}r=16(1-\lambda ){\u03f5}_{1}.\end{array}$$

In terms of the potential, the slow-roll parameters are expressed as [70]

$${\u03f5}_{1}=\frac{1-\lambda}{2}{\left(\frac{{V}^{\prime}}{V}\right)}^{2},$$

$${\u03f5}_{2}=-2(1-\lambda )\left[\frac{{V}^{\u2033}}{V}-{\left(\frac{{V}^{\prime}}{V}\right)}^{2}\right].$$

Due to the factor $1-\lambda $ in Equations (10) and (11), even if the gradient of the potential is consistent with $\mathcal{SC}$II, the slow-roll conditions (6) can still be satisfied as long as $\lambda $ is close to 1, so the slow-roll results (8) and (9) are applicable, and the e-folding number N can reach $N=60$ easily. Substituting Equation (10) into Equation (9), we get

$$r=8{(1-\lambda )}^{2}{\left(\frac{{V}^{\prime}}{V}\right)}^{2}.$$

From Equation (12), we see that while condition (2) is satisfied, the tensor-to-scalar ratio r can still be very small, as long as $1-\lambda $ is small enough. In particular, if we combine the observational constraint ${r}_{0.002}<0.064$ with the condition (2), we get

$$1-\lambda <\frac{0.09}{c}.$$

Now we discuss the first swampland criterion $\mathcal{SC}$I for the field excursion. The Lyth bound tells us that [15,49]

$$\Delta \varphi >\Delta N\sqrt{\frac{r}{8}}=(1-\lambda )\Delta N\frac{|{V}^{\prime}|}{V}.$$

Without the Gauss–Bonnet term, i.e., $\lambda =0$, if $\mathcal{SC}$II is satisfied, then it is impossible to satisfy $\mathcal{SC}$I for single-field slow-roll inflation with $\Delta N\sim 60$. With the help the Gauss–Bonnet term, it is very easy to satisfy both $\mathcal{SC}$I and $\mathcal{SC}$II conditions, as long as $1-\lambda $ is small enough. Combining Equations (1) and (2), we obtain the constraint

$$1-\lambda <\frac{d}{c\Delta N}.$$

In summary, with the help of the Gauss–Bonnet term, as long as the order one parameter $\lambda $ satisfies Equations (13) and (15), the two swampland criteria $\mathcal{SC}$I and $\mathcal{SC}$II are satisfied, and the tensor-to-scalar ratio r is also consistent with the observations [12]. From Equation (8), we see that the parameter $\lambda $ have no effect on the scalar spectral tilt ${n}_{s}$, so the constraint on ${n}_{s}$ can also be satisfied.

In the next section, we will use two inflationary models, the power-law potential and the E-model, as examples to support the above discussion.

## 3. The Models

In the following, we consider two inflationary models, the chaotic inflation and the E-model. We show that with the help of the Gauss–Bonnet term, the two swampland criteria $\mathcal{SC}$I and $\mathcal{SC}$II are satisfied for both models. Additionally, the models also satisfy the observational constraints [12],

$${n}_{s}=0.9649\pm 0.0042,\phantom{\rule{1.em}{0ex}}{r}_{0.002}<0.064.$$

#### 3.1. The Power-Law Potential

For the chaotic inflation with the power-law potential [71]
the excursion of the inflaton is
where N is the remaining number of e-folds before the end of inflation, and

$$V={V}_{0}{\varphi}^{p},$$

$$\Delta \varphi =\sqrt{2(1-\lambda )p}\left(\sqrt{N+\tilde{n}}-\sqrt{\tilde{n}}\right),$$

$$\tilde{n}=\left\{\begin{array}{cc}p/4,& 0<p<2,\\ (p-1)/2,& p\ge 2.\end{array}\right.$$

The scalar spectral tilt ${n}_{s}$ and the tensor-to-scalar ratio r are

$$\begin{array}{c}{n}_{s}-1=-\frac{p+2}{2(N+\tilde{n})},\end{array}$$

$$\begin{array}{c}r=\frac{4(1-\lambda )p}{N+\tilde{n}}.\end{array}$$

From Equation (20), we see that ${n}_{s}$ is independent on $\lambda $. The slow-roll parameter ${\eta}_{V}>0$ if $p>1$; and ${\eta}_{V}<0$ if $p<1$, so the condition (3) cannot be satisfied for the chaotic model with $p>1$. If we choose $p=2$ and $N=60$, the scalar spectral tilt is ${n}_{s}=0.9669$, which is consistent with the observations (16). Although condition (3) is violated, condition (2) can be satisfied. By varying the value of $\lambda $, the values of the gradient of the potential ${V}^{\prime}/V$, inflaton excursion $\Delta \varphi $, and tensor-to-scalar ratio r are shown in Figure 1.

As the value of $1-\lambda $ becomes smaller and smaller, r and $\Delta \varphi $ become smaller and smaller too, but ${V}^{\prime}/V$ will become larger and larger. If $1-\lambda $ satisfies the conditions (13) and (15), the swampland criteria (1) and (2) as well as the observational constraints (16) are satisfied. For example, if we choose $1-\lambda =2\times {10}^{-3}$, we get

$$\begin{array}{c}{n}_{s}=0.9669,\phantom{\rule{1.em}{0ex}}r=2.6\times {10}^{-4},\end{array}$$

$$\begin{array}{c}\Delta \varphi =0.63,\phantom{\rule{1.em}{0ex}}\frac{{V}^{\prime}}{V}=2.9.\end{array}$$

#### 3.2. The E-Model

For the E-model [73,74]
the excursion of the inflaton for $n=1$ is
where

$$V={V}_{0}{\left[1-exp\left(-\sqrt{\frac{2}{3\alpha}}\varphi \right)\right]}^{2n},$$

$$\Delta \varphi =-\frac{1}{\sqrt{6\alpha}}\left[3\alpha (\tilde{N}+X-1)+4(1-\lambda )N\right],$$

$$\tilde{N}=1+{W}_{-1}\left[-Xexp\left(-X-\frac{4(1-\lambda )N}{3\alpha}\right)\right],$$

$X=2\sqrt{1-\lambda}/\left(\sqrt{3\alpha}\right)+1$, and the function ${W}_{-1}$ is the lower branch of the Lambert W function. The scalar spectral tilt ${n}_{s}$ and the tensor-to-scalar ratio r are [75]

$$\begin{array}{c}{n}_{s}=1+\frac{8(1-\lambda )}{3\alpha \tilde{N}}-\frac{16(1-\lambda )}{3\alpha {\tilde{N}}^{2}},\end{array}$$

$$\begin{array}{c}r=\frac{64{(1-\lambda )}^{2}}{3\alpha {\tilde{N}}^{2}}.\end{array}$$

If we choose $\alpha =1-\lambda $ and $N=60$, we get ${n}_{s}=0.9678$, which is consistent with the observation. Varying $\lambda $, the values of ${V}^{\u2033}/V$, ${V}^{\prime}/V$, $\Delta \varphi $, and r are shown in Figure 2.

Similar to the chaotic inflation, as the value of $1-\lambda $ becomes smaller and smaller, r and $\Delta \varphi $ become smaller and smaller and $-{V}^{\u2033}/V$ and ${V}^{\prime}/V$ become larger and larger. When $1-\lambda $ satisfies the conditions (13) and (15), all the swampland criteria as well as the observational constraints (16) are satisfied. If we choose $1-\lambda =2\times {10}^{-3}$, we get

$$\begin{array}{c}{n}_{s}=0.9678,\phantom{\rule{1.em}{0ex}}r=5.9\times {10}^{-6},\end{array}$$

$$\begin{array}{c}\Delta \varphi =0.20,\phantom{\rule{1.em}{0ex}}\frac{{V}^{\u2033}}{V}=-7.8.\end{array}$$

The predictions (29) are consistent with the observations (16), and the swampland criteria $\mathcal{SC}$I (1) and $\mathcal{SC}$II (2) are satisfied. Further more, if we choose $1-\lambda ={10}^{-4}$, we get

$$\begin{array}{c}{n}_{s}=0.9678,\phantom{\rule{1.em}{0ex}}r=3.0\times {10}^{-7},\end{array}$$

$$\begin{array}{c}\Delta \varphi =0.045,\phantom{\rule{1.em}{0ex}}\frac{{V}^{\prime}}{V}=1.9,\phantom{\rule{1.em}{0ex}}\frac{{V}^{\u2033}}{V}=-155.3.\end{array}$$

## 4. Conclusions

The two swampland criteria pose a threat on single-field slow-roll inflation. With the help of the Gauss–Bonnet coupling, the relationship between r and ${V}^{\prime}/V$ is described by Equation (12), i.e., r is reduced by a factor of ${(1-\lambda )}^{2}$ compared with the result in standard single-field slow-roll inflation. Due to the reduction in r, the first swampland criterion is easily satisfied by requiring r to be small. On the other hand, it is easy to satisfy the second swampland criterion by requiring $1-\lambda $ to be small and keeping r small.

For the chaotic inflation with $p=2$, if we take $1-\lambda =5\times {10}^{-5}$, we get ${n}_{s}=0.9669$, $r=6.6\times {10}^{-6}$, ${V}^{\prime}/V=18.2$, and $\Delta \varphi =0.1$. Therefore, for the chaotic inflation with $p=2$ and $\lambda >0.99995$, the model satisfies not only the observational constraints, but also the swampland criteria (1) and (2). If $p<1$, although condition (3) can be satisfied, the model is inconsistent with the observation at the $1\sigma $ confidence level. For the E-model, if we choose $1-\lambda ={10}^{-4}$, we get ${n}_{s}=0.9678$, $r=3.0\times {10}^{-7}$, ${V}^{\prime}/V=1.9$, ${V}^{\u2033}/V=-155.3$, and $\Delta \varphi =0.045$. The model satisfies not only the observational constraints, but also the swampland criteria (1), (2), and (3). In conclusion, the Gauss–Bonnet inflation with the condition (5) satisfies not only the observational constraints, but also the swampland criteria. Therefore, the Gauss–Bonnet coupling seems a way out of the swampland issue for single-field inflationary models.

## Author Contributions

Conceptualization, Y.G.; investigation, Z.Y.; data curation, Z.Y.; writing—original draft preparation, Z.Y.; writing—review and editing, Y.G.; supervision, Y.G.; funding acquisition, Y.G.

## Funding

This research was funded in part by the National Natural Science Foundation of China under Grant Nos. 11875136 and 11475065 and the Major Program of the National Natural Science Foundation of China under Grant No. 11690021.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

- Guth, A.H. The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Phys. Rev. D
**1981**, 23, 347–356. [Google Scholar] [CrossRef] - Linde, A.D. A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems. Phys. Lett. B
**1982**, 108, 389–393. [Google Scholar] [CrossRef] - Albrecht, A.; Steinhardt, P.J. Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking. Phys. Rev. Lett.
**1982**, 48, 1220–1223. [Google Scholar] [CrossRef] - Starobinsky, A.A. A New Type of Isotropic Cosmological Models Without Singularity. Phys. Lett. B
**1980**, 91, 99–102. [Google Scholar] [CrossRef] - Sato, K. First Order Phase Transition of a Vacuum and Expansion of the Universe. Mon. Not. R. Astron. Soc.
**1981**, 195, 467–479. [Google Scholar] [CrossRef] - Ooguri, H.; Vafa, C. On the Geometry of the String Landscape and the Swampland. Nucl. Phys. B
**2007**, 766, 21–33. [Google Scholar] [CrossRef] - Vafa, C. The String landscape and the swampland. arXiv
**2005**, arXiv:hep-th/0509212. [Google Scholar] - Ooguri, H.; Vafa, C. Non-supersymmetric AdS and the Swampland. Adv. Theor. Math. Phys.
**2017**, 21, 1787–1801. [Google Scholar] [CrossRef] - Obied, G.; Ooguri, H.; Spodyneiko, L.; Vafa, C. De Sitter Space and the Swampland. arXiv
**2018**, arXiv:hep-th/1806.08362. [Google Scholar] - Ooguri, H.; Palti, E.; Shiu, G.; Vafa, C. Distance and de Sitter Conjectures on the Swampland. Phys. Lett. B
**2019**, 788, 180–184. [Google Scholar] [CrossRef] - Kehagias, A.; Riotto, A. A note on Inflation and the Swampland. arXiv
**2018**, arXiv:hep-th/1807.05445. [Google Scholar] [CrossRef] - Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. Planck 2018 results. X. Constraints on inflation. arXiv
**2018**, arXiv:1807.06211. [Google Scholar] - Ade, P.A.R.; Ahmed, Z.; Aikin, R.W.; Alexander, K.D.; Barkats, D.; Benton, S.J.; Bischoff, C.A.; Bock, J.J.; Bowens-Rubin, R.; Brevik, J.A.; et al. BICEP2 / Keck Array x: Constraints on Primordial Gravitational Waves using Planck, WMAP, and New BICEP2/Keck Observations through the 2015 Season. Phys. Rev. Lett.
**2018**, 121, 221301. [Google Scholar] [CrossRef] - Agrawal, P.; Obied, G.; Steinhardt, P.J.; Vafa, C. On the Cosmological Implications of the String Swampland. Phys. Lett. B
**2018**, 784, 271–276. [Google Scholar] [CrossRef] - Lyth, D.H. What would we learn by detecting a gravitational wave signal in the cosmic microwave background anisotropy? Phys. Rev. Lett.
**1997**, 78, 1861–1863. [Google Scholar] [CrossRef] - Berera, A. Warm inflation. Phys. Rev. Lett.
**1995**, 75, 3218–3221. [Google Scholar] [CrossRef] - Brennan, T.D.; Carta, F.; Vafa, C. The String Landscape, the Swampland, and the Missing Corner. arXiv
**2017**, arXiv:1711.00864. [Google Scholar] - Das, S. Note on single-field inflation and the swampland criteria. Phys. Rev. D
**2019**, 99, 083510. [Google Scholar] [CrossRef] - Das, S. Warm Inflation in the light of Swampland Criteria. Phys. Rev. D
**2019**, 99, 063514. [Google Scholar] [CrossRef] - Motaharfar, M.; Kamali, V.; Ramos, R.O. Warm inflation as a way out of the swampland. Phys. Rev. D
**2019**, 99, 063513. [Google Scholar] [CrossRef] - Ashoorioon, A. Rescuing Single Field Inflation from the Swampland. Phys. Lett. B
**2019**, 790, 568–573. [Google Scholar] [CrossRef] - Lin, C.M.; Ng, K.W.; Cheung, K. Chaotic inflation on the brane and the Swampland Criteria. Phys. Rev. D
**2019**, 100, 023545. [Google Scholar] [CrossRef] - Lin, C.M. Type I Hilltop Inflation and the Refined Swampland Criteria. Phys. Rev. D
**2019**, 99, 023519. [Google Scholar] [CrossRef] - Kinney, W.H.; Vagnozzi, S.; Visinelli, L. The zoo plot meets the swampland: Mutual (in)consistency of single-field inflation, string conjectures, and cosmological data. Class. Quant. Grav.
**2019**, 36, 117001. [Google Scholar] [CrossRef] - Achúcarro, A.; Palma, G.A. The string swampland constraints require multi-field inflation. J. Cosmol. Astropart. Phys.
**2019**, 2019, 041. [Google Scholar] [CrossRef] - Andriot, D. On the de Sitter swampland criterion. Phys. Lett. B
**2018**, 785, 570–573. [Google Scholar] [CrossRef] - Park, S.C. Minimal gauge inflation and the refined Swampland conjecture. J. Cosmol. Astropart. Phys.
**2019**, 2019, 053. [Google Scholar] [CrossRef] - Garg, S.K.; Krishnan, C. Bounds on Slow Roll and the de Sitter Swampland. arXiv
**2018**, arXiv:1807.05193. [Google Scholar] - Garg, S.K.; Krishnan, C.; Zaid Zaz, M. Bounds on Slow Roll at the Boundary of the Landscape. J. High Energy Phys.
**2019**, 2019, 029. [Google Scholar] [CrossRef] - Schimmrigk, R. The Swampland Spectrum Conjecture in Inflation. arXiv
**2018**, arXiv:1810.11699. [Google Scholar] - Dimopoulos, K. Steep Eternal Inflation and the Swampland. Phys. Rev. D
**2018**, 98, 123516. [Google Scholar] [CrossRef] - Matsui, H.; Takahashi, F. Eternal Inflation and Swampland Conjectures. Phys. Rev. D
**2019**, 99, 023533. [Google Scholar] [CrossRef] - Ben-Dayan, I. Draining the Swampland. Phys. Rev. D
**2019**, 99, 101301. [Google Scholar] [CrossRef] - Brahma, S.; Wali Hossain, M. Avoiding the string swampland in single-field inflation: Excited initial states. J. High Energy Phys.
**2019**, 2019, 006. [Google Scholar] [CrossRef] - Roupec, C.; Wrase, T. De Sitter Extrema and the Swampland. Fortsch. Phys.
**2019**, 67, 1800082. [Google Scholar] [CrossRef] - Blåbäck, J.; Danielsson, U.; Dibitetto, G. A new light on the darkest corner of the landscape. arXiv
**2018**, arXiv:1810.11365. [Google Scholar] - Odintsov, S.D.; Oikonomou, V.K. Finite-time Singularities in Swampland-related Dark Energy Models. Europhys. Lett.
**2019**, 126, 20002. [Google Scholar] [CrossRef] - Kawasaki, M.; Takhistov, V. Primordial Black Holes and the String Swampland. Phys. Rev. D
**2018**, 98, 123514. [Google Scholar] [CrossRef] - Wang, S.J. Electroweak relaxation of cosmological hierarchy. Phys. Rev. D
**2019**, 99, 023529. [Google Scholar] [CrossRef] - Heisenberg, L.; Bartelmann, M.; Brandenberger, R.; Refregier, A. Dark Energy in the Swampland. Phys. Rev. D
**2018**, 98, 123502. [Google Scholar] [CrossRef] - Agrawal, P.; Obied, G. Dark Energy and the Refined de Sitter Conjecture. J. High Energy Phys.
**2019**, 2019, 103. [Google Scholar] [CrossRef] - Dvali, G.; Gomez, C.; Zell, S. Quantum Breaking Bound on de Sitter and Swampland. Fortsch. Phys.
**2019**, 67, 1800094. [Google Scholar] [CrossRef] - Fukuda, H.; Saito, R.; Shirai, S.; Yamazaki, M. Phenomenological Consequences of the Refined Swampland Conjecture. Phys. Rev. D
**2019**, 99, 083520. [Google Scholar] [CrossRef] - Chiang, C.I.; Leedom, J.M.; Murayama, H. What does inflation say about dark energy given the swampland conjectures? Phys. Rev. D
**2019**, D100, 043505. [Google Scholar] [CrossRef] - Sabir, M.; Ahmed, W.; Gong, Y.; Lu, Y. Superconformal attractor E-models in brane inflation under swampland criteria. arXiv
**2019**, arXiv:1903.08435. [Google Scholar] - Channuie, P. Refined Swampland conjecture in deformed Starobinsky gravity. arXiv
**2019**, arXiv:1907.10605. [Google Scholar] - Palti, E. The Swampland: Introduction and Review. Fortsch. Phys.
**2019**, 67, 1900037. [Google Scholar] [CrossRef] - Kamali, V. Reheating After Swampland Conjecture. arXiv
**2019**, arXiv:1902.00701. [Google Scholar] - Yi, Z.; Gong, Y.; Sabir, M. Inflation with Gauss-Bonnet coupling. Phys. Rev. D
**2018**, 98, 083521. [Google Scholar] [CrossRef] - Antoniadis, I.; Rizos, J.; Tamvakis, K. Singularity—Free cosmological solutions of the superstring effective action. Nucl. Phys. B
**1994**, 415, 497–514. [Google Scholar] [CrossRef] - Kawai, S.; Sakagami, M.A.; Soda, J. Instability of one loop superstring cosmology. Phys. Lett. B
**1998**, 437, 284–290. [Google Scholar] [CrossRef] - Kawai, S.; Soda, J. Evolution of fluctuations during graceful exit in string cosmology. Phys. Lett. B
**1999**, 460, 41–46. [Google Scholar] [CrossRef] - Tsujikawa, S. Density perturbations in the ekpyrotic universe and string inspired generalizations. Phys. Lett. B
**2002**, 526, 179–185. [Google Scholar] [CrossRef] - Toporensky, A.; Tsujikawa, S. Nature of singularities in anisotropic string cosmology. Phys. Rev. D
**2002**, 65, 123509. [Google Scholar] [CrossRef] - Gong, Y.; Papantonopoulos, E.; Yi, Z. Constraints on scalar–tensor theory of gravity by the recent observational results on gravitational waves. Eur. Phys. J. C
**2018**, 78, 738. [Google Scholar] [CrossRef] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A. Astrophys. J.
**2017**, 848, L13. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K. Inflationary Phenomenology of Einstein Gauss-Bonnet Gravity Compatible with GW170817. arXiv
**2019**, arXiv:1908.07555. [Google Scholar] [CrossRef] - Rizos, J.; Tamvakis, K. On the existence of singularity free solutions in quadratic gravity. Phys. Lett. B
**1994**, 326, 57–61. [Google Scholar] [CrossRef] - Kanti, P.; Rizos, J.; Tamvakis, K. Singularity free cosmological solutions in quadratic gravity. Phys. Rev. D
**1999**, 59, 083512. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D.; Sasaki, M. Gauss-Bonnet dark energy. Phys. Rev. D
**2005**, 71, 123509. [Google Scholar] [CrossRef] - Satoh, M.; Soda, J. Higher Curvature Corrections to Primordial Fluctuations in Slow-roll Inflation. J. Cosmol. Astropart. Phys.
**2008**, 2008, 019. [Google Scholar] [CrossRef] - Kanti, P.; Gannouji, R.; Dadhich, N. Gauss-Bonnet Inflation. Phys. Rev. D
**2015**, 92, 041302. [Google Scholar] [CrossRef] - Astashenok, A.V.; Odintsov, S.D.; Oikonomou, V.K. Modified Gauss–Bonnet gravity with the Lagrange multiplier constraint as mimetic theory. Class. Quant. Grav.
**2015**, 32, 185007. [Google Scholar] [CrossRef] - Chakraborty, S.; Paul, T.; SenGupta, S. Inflation driven by Einstein-Gauss-Bonnet gravity. Phys. Rev. D
**2018**, 98, 083539. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K. Viable Inflation in Scalar-Gauss-Bonnet Gravity and Reconstruction from Observational Indices. Phys. Rev. D
**2018**, 98, 044039. [Google Scholar] [CrossRef] - Chatzarakis, N.; Oikonomou, V.K. Autonomous Dynamical System of Einstein-Gauss-Bonnet Cosmologies. arXiv
**2019**, arXiv:1908.08141. [Google Scholar] - Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution. Phys. Rept.
**2017**, 692, 1–104. [Google Scholar] [CrossRef] - Van de Bruck, C.; Dimopoulos, K.; Longden, C. Reheating in Gauss-Bonnet-coupled inflation. Phys. Rev. D
**2016**, 94, 023506. [Google Scholar] [CrossRef] - Schwarz, D.J.; Terrero-Escalante, C.A.; Garcia, A.A. Higher order corrections to primordial spectra from cosmological inflation. Phys. Lett. B
**2001**, 517, 243–249. [Google Scholar] [CrossRef] - Guo, Z.K.; Schwarz, D.J. Slow-roll inflation with a Gauss-Bonnet correction. Phys. Rev. D
**2010**, 81, 123520. [Google Scholar] [CrossRef] - Linde, A.D. Chaotic Inflation. Phys. Lett. B
**1983**, 129, 177–181. [Google Scholar] [CrossRef] - Gerbino, M.; Freese, K.; Vagnozzi, S.; Lattanzi, M.; Mena, O.; Giusarma, E.; Ho, S. Impact of neutrino properties on the estimation of inflationary parameters from current and future observations. Phys. Rev.
**2017**, D95, 043512. [Google Scholar] [CrossRef] - Kallosh, R.; Linde, A. Non-minimal Inflationary Attractors. JCAP
**2013**, 2013, 033. [Google Scholar] [CrossRef] - Carrasco, J.J.M.; Kallosh, R.; Linde, A. Cosmological Attractors and Initial Conditions for Inflation. Phys. Rev. D
**2015**, 92, 063519. [Google Scholar] [CrossRef] - Yi, Z.; Gong, Y. Nonminimal coupling and inflationary attractors. Phys. Rev. D
**2016**, 94, 103527. [Google Scholar] [CrossRef]

**Figure 1.**The dependence on $\lambda $ for the power-law potential with $p=2$. The upper panel shows the tensor-to-scalar r. The lower panel shows the gradient of the potential ${V}^{\prime}/V$ and the field excursion $\Delta \varphi $.

**Figure 2.**The dependence on $\lambda $ for the the E-model with $\alpha =1-\lambda $. The upper panel shows the tensor-to-scalar r. The lower panel shows the gradients of the potential ${V}^{\u2033}/V$ and ${V}^{\prime}/V$, and the field excursion $\Delta \varphi $.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).